Fibrations associated
with
$*$-homomorphisms
of
$\mathrm{C}^{*}$-algebras
京大数理研
川村勝紀
(Katsunori Kawamura)
1
Introduction
We consider geometric counter parts of$\mathrm{C}^{*}$-algebras. By [1] and our studies, we have
the following tabular of correspondences as a generalization of classical
(commuta-tive) known results: $-$
where the part of the group of automorphisms and modules are our studies $[2, 3]$.
We want to fill ? in this tabular. $\mathrm{C}^{*}$-subalgebra and representation are treated
as special studies of homomorphisms between two $\mathrm{C}^{*}$-algebras.
The aim of present paper is a presentation of studies of fibration structure
as-sociated $\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}*$-homomorphisms for two unital $\mathrm{C}^{*}$-algebras $A$ and $B$. We introduce three levels of studies as follows:
(i) one homomorphism $\phi\in \mathrm{H}\mathrm{o}\mathrm{m}(A, B)$ itself,
(ii) the space $\mathrm{H}\mathrm{o}\mathrm{m}(A,B)$ of$\mathrm{a}\mathrm{l}1*$-homomorphisms, (iii) a $\mathrm{b}\mathrm{i}$-functor $\mathrm{H}\mathrm{o}\mathrm{m}$
.
We introduce examples of three levels (i), (ii), (iii). We show relations between
2
General
setting
In [1],
Theorem 2.1 Each unital $C^{*}$-algebra $A$ can be realized faithfully as a
function
sub-algebra on the $set/p$
of
allpure states on it asGelfand
representation$f$ : $Aarrow \mathcal{F}(\mathcal{P})$; $A\vdash+f_{A}$, $f_{A}(\rho)\equiv\rho(A)$ $(\rho\in \mathcal{P})$
with $a*$-product on the
function
space $\mathcal{F}(P)$ on $P$.This theorem brings us a possibility of a geometrical study of $\mathrm{C}^{*}$-algebra. Though
the correspondence between a $\mathrm{C}^{*}$-algebra and $P$ is not categorical, that is, it is not
suitable for studies of morphisms and subalgebras. Hence we try to consider the
following two possibilities:
(i) give up to study about their categorical property and study the phenomena
which appear from morphisms and subalgebras.
(ii) find the other generalization of the correspondence between $\mathrm{C}^{*}$-algebras and
its dual objects.
We review facts in commutative case once more. Let $X$ and $\mathrm{Y}$ be two compact
Hausdorffspaces. Then there is a $\mathrm{o}\mathrm{n}\mathrm{e}- \mathrm{t}\mathrm{o}$-one correspondence between $C(X, Y)$ and
$\mathrm{H}\mathrm{o}\mathrm{m}(C(\mathrm{Y}), c(X))$. For example, $f$
:
$Xarrow Y$ is surjective if and only if $f^{*}$ :$C(Y)arrow C(X)$ is injective. In this sense, subalgebra of commutative $\mathrm{C}^{*}$-algebra
is corresponded to a $\mathrm{f}\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\Gamma_{\wedge}$. Then there is a question what happens in the non
commutative case.
In general, for $\phi\in \mathrm{H}\mathrm{o}\mathrm{m}(A, \beta),$ $\emptyset*(\mathcal{P}_{B})\not\subset P_{A}$
.
This is one of problems of breakdownof the categorical equivalence in the commutative case. We treat this type of dual
of morphisms in the next section.
3
A fibration associated with single morphism
As the case (i) in section 1, we explain easy examples. We obtain the following
fibration from an inclusion $\iota$ : $\mathrm{C}^{2}rightarrow M_{2}(\mathrm{C})$ as an observation of a naive spin
Proposition 3.1 (Torus
fibration for
$S^{3}$) There is a continuous surjection$\mu$ : $S^{3}arrow$$[0,1]$
of
$S^{3}$ on a closed interval $[0,1]$ and thisfibration
induces a decomposition asfollows:
$S^{3} \cong\prod_{z\in[0,1]}T_{z}$
where $T_{0}\cong\tau_{1}\cong S^{1}$ and $T_{z}\cong T^{2}$
for
$z\in(\mathrm{O}, 1)$.Proof.
Define a map$f$ : $s^{3}=s(\mathrm{c}^{2})arrow[0,1]$
by
$f(Z_{1}, z_{2})\equiv|z_{1}|$ $((z_{1,2}z)\in S^{3})$.
Then $f$ is surjective and continuous. The fibration by $f$ gives the statement of
proposition. 1
Specially, a map $f’$ which is induced by $f$
$f^{J}$ : $\mathrm{C}P^{1}\equiv S^{3}/U(1)arrow[0,1]$
is just the restriction of dual map $\iota$ : $\mathrm{C}^{2}\mathrm{c}arrow M_{2}(\mathrm{C})$ to the set
$\mathrm{C}P^{1}$ of pure states
of $M_{2}(\mathrm{C})$. $[0,1]$ is the set of all(mixed)states of
$\mathrm{C}^{2}$. The fiber of $f^{J}$ is regarded as
lost freedom in the observation of quantum states of $M_{2}(\mathrm{C})$. The image of$f^{J}$ is the
probability which the spin takes value, $0$($=$ down) or 1$(=\mathrm{u}\mathrm{p})$
.
In this sense, wenote that $[0,1]$ is a l-simplex.
In other point of view, Proposition 3.1 represents the locus of deformation of
torus $T^{2}$ in $\mathrm{C}^{2}\cong \mathrm{R}^{4}$. Or, $S^{3}$ can be considered as a locus of$T^{2}$ in $\mathrm{R}^{4}$ which moves
with 1-parameter. At point of the start and the end in $[0,1],$ $T^{2}$ collapses $S^{1}$ by
pinching out one of two cycles of $T^{2}$.
In the same way, we have a little bit general result.
Proposition 3.2 Any $2n+1$-dimensional sphere has a singular
fibration
on an-simplex with $k+1$-torus as a
fiber
at a point on the interiorof
k-subsimplex.In general, the normal (mixed)state space of $\mathcal{L}(\mathcal{H})$ has a fibration on a Hilbert
simplex with flag manifold as the fiber. This is proved by the uniqueness of
diago-nalizationof positive normalizedtrace class operator as its ordered eigen values for a
completeorthonormal basis. In this way, the state space of an algebra ofobservables
has a fibration with respect to the spectrum of the commutative algebra affiliated
with Hamiltonian. This is just an observation in a quantum system. We can watch
lost freedom in an observation as a flag manifold.
4
Fibrations
of
homomorphism spaces
$i$
4.1
Morphisms between
matrix
algebras
We show easy examples as the case (ii) in section 1. We consider $H_{2,3}$ defined by
the set of$\mathrm{a}\mathrm{l}1*$-homomorphisms:
The content of$H_{2,3}$ is classified two types, $\phi=0$ or not. Define$\Lambda_{2,3}\equiv H_{2,3}/\sim \mathrm{w}\mathrm{h}\mathrm{e}x\mathrm{e}$
$\phi\sim\phi’$ if there is $\mathrm{a}*$-automorphism
$g$ of $M_{2}(\mathrm{C})$ such that $\phi’=\phi \mathrm{o}g$
.
Remark thatthe automorphism group of $M_{2}(\mathrm{C})$ is isomorphic to the projective unitary group
PU(2) $\equiv U(2)/U(1)$
.
Denote the natural projection$\pi$ : $H_{2,3}arrow\Lambda_{2,3}$
.
Then we have a fibration $(H_{2,3}, \pi, \Lambda_{2,3})$. We have the following proposition
Proposition 4.1 There is the following equivalence
of
fibration:
$(H_{2,3}, \pi, \Lambda_{2,3})\cong(PV_{2}(\mathrm{C}^{3})\cup\{0\}, 1^{\text{ノ}}\sim, \mathrm{c}P^{2}\cup\{0\})$
where $(PV_{2}(\mathrm{C}^{3}), \iota \text{ノ}, \mathrm{c}P^{2})$ is a principal PU(2)-bundle, $PV_{2}(\mathrm{c}^{3})$ is projective
Stiefel
manifold
consistingof
$U(1)$-orbitsof
2-orthonormal basisof
$\mathrm{C}^{3}$ and $\tilde{\nu}$ is theexten-sion
of
lノ by $\tilde{\nu}(0)=0$.
Proof.
For $0\neq\phi\in H_{2,3}$, the image $\phi(I_{2})$ of the unit $I_{2}$ of $M_{2}(\mathrm{C})$ is a2-dimensional
projection on $\mathrm{C}^{3}$
.
On the other hand, for a twodimensional
subspace $V\subset \mathrm{C}^{3}$, we
obtain a isomorphism $\mathrm{E}\mathrm{n}\mathrm{d}_{\mathrm{C}}(V)\cong M_{2}(\mathrm{c})$ up to automorphismof$M_{2}(\mathrm{C})$
.
Hencewehave an equivalence
$\Lambda_{2,3}^{\cross}\equiv$ $\{\phi\in H_{2,3} : \phi\neq 0\}/\sim$
$\cong$
$\{\phi(I_{2}):\emptyset\in H2,3\backslash \{0\}\}$ $\cong$ $G_{2}(\mathrm{c}^{\mathrm{s}})$
$\cong$ $\mathrm{C}P^{2}$
.
For $\phi,$$\phi^{l}\in H_{2,3}^{\cross}$ such that $\phi\sim\phi^{J}$, then $\phi^{-1}0\phi’\in \mathrm{A}\mathrm{u}\mathrm{t}M_{2}(\mathrm{C})\cong PU(2)$
.
The actionof PU(2) on the fiber on $0\neq z\in\Lambda_{2,3}$ is free. Hence
$\pi^{-1}(z)\cong PU(2)$ $(0\neq z\in\Lambda_{2,3})$
.
The correspondence between $H_{2,3}^{\cross}$ and $PV_{2}(\mathrm{C}^{3})$ is given to calculate the isotropy
subgroup of PU (3)-(left)action on $H_{2,3}\backslash \{0\}$
.
In the same way, we have
Proposition 4.2 There is an equivalence
of
fibrations:
$(H_{2,4}, \pi, \Lambda_{2,4})$
$\cong(\{0\}\cup PV_{2}(\mathrm{C}^{4})\cup FM_{2}(\mathrm{C}^{4}),\tilde{\nu}, \{0\}\cup G_{2}(\mathrm{C}^{4})\cup G_{2}(\mathrm{C}^{4})/\mathrm{Z}_{2})$
where $FM_{2}(\mathrm{C}^{4})$ is a homogeneous space
defined
by$FM_{2}(\mathrm{C}^{4})\equiv PU(4)/P(U(2)\otimes I_{2})$
.
The
fiber of
thisfibration
is equal to PU(2) except $\{0\}$.$FM_{2}(\mathrm{C}^{4})$ is the space of all fermions on $\mathrm{C}^{4}$ because an element of
$FM_{2}(\mathrm{C}^{4})$ is
a homomorphism determined by a partial isometry $v$ on $\mathrm{C}^{4}$ which satisfies the
canonical anti-commutation relation:
where $\{\cdot, \cdot\}$ is the anti-commutator on $M_{4}(\mathrm{C})$. $FM_{2}(\mathrm{C}^{4})$ has no name in usual text
book of geometry and it is not known other realization like a
Stiefel manifold
yet.For $n,$$m\in \mathrm{N},$ $H_{n,m}$ is described as a union of space of $n-1$-chain of
B-construction for the category of projections on $\mathrm{C}^{m}$. For example, an element of
$FM_{2}(\mathrm{C}^{4})$ is a 1-chain, that is an edge between two orthogonal $\mathrm{p}\mathrm{r}\mathrm{o}.\mathrm{j}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$ on
$\mathrm{C}^{4}$.
$0\neq\phi\in H_{3,6}$ corresponds to $2$-chain(,$\mathrm{o}\mathrm{r}2$-simplex) of the set of
projections
on $\mathrm{C}^{6}$.
A matrix algebra has one-point spectrum. Hence it is
regarded
as a point withsome kind of internal freedom. $H_{2,3}$ is a non
commutative
mapping space betweennon commutative points with internal symmetries PU (2) and PU(3), respectively.
4.2
Structure
theorem of
representation
space
of
Cuntz
al-gebra
Let $\mathcal{O}_{n}$ be a Cuntz algebra with Cuntz
generator
$\{s_{i}\}_{i=1}^{n},$ $n\geq 2$, and$\mathcal{H}$ a
sepa-rable infinite
dimensional
Hilbert space. Denote Rep$(\mathcal{O}n’ \mathcal{H})$ the set of all unital$*$
-representations
of $\mathcal{O}_{n}$ on $\mathcal{H}$.We show that Rep$(\mathcal{O}_{n}, \mathcal{H})$ is realized as a fiber product of
representation
spaceof matrix algebra and space of some partial isometries.
Remark
Rep$(\mathcal{O}n’ \mathcal{H})=$$\mathrm{H}\mathrm{o}\mathrm{m}(\mathcal{O}_{n}, \mathcal{L}(\mathcal{H}))$.
Let $G_{\infty,\infty}(\mathcal{H})$ be the set of all projections on
$\mathcal{H}$ with both rank and $\mathrm{c}\mathrm{o}$-rank $\infty$. We call $G_{\infty,\infty}(\mathcal{H})$ the Grassmanian on $\mathcal{H}$ type
of
$(\infty, \infty)$. Let $H_{n}(\mathcal{H})$ be the set ofall $\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{l}*$-homomorphisms from $M_{n}(\mathrm{C})$ to $\mathcal{L}(H)$. Let $PI(I, G_{\infty,\infty}(\mathcal{H}))$ be the set
of all partial isometries consisting $v$ satisfying
$\iota$
$v^{*}v=I$, $vv^{*}\in G_{\infty,\infty}(\mathcal{H})$.
Then we have three fibrations on $G_{\infty,\infty}(\mathcal{H})$ as follows:
$\pi$ : Rep$(\mathcal{O}_{n}, \mathcal{H})$ $arrow$ $G_{\infty,\infty}(\mathcal{H})$,
$\nu$ : $H_{n}(\mathcal{H})$ $arrow$ $G_{\infty,\infty}(\mathcal{H})$,
$\mu$ : $PI(I, G_{\infty,\infty}(\mathcal{H}))$ $arrow$
$G_{\infty,\infty}(\mathcal{H})$
defined by
$\pi(\phi)\equiv$ $\phi(s_{1}s_{1}*)$ $(\phi\in \mathrm{R}\mathrm{e}\mathrm{p}(\mathcal{O}_{n}, \mathcal{H}))$,
$\nu(\psi)\equiv$ $\psi(E_{11})$ $(\psi_{\in}H_{n}(\mathcal{H}))$,
$\mu(v)\equiv$ $vv^{*}$ $(v\in PI(I, G\infty,\infty(\mathcal{H})))$
where $\{E_{ij}\}^{n}i,j=1$ is the canonical matrix unit of$M_{n}(\mathrm{C})$. Define the fiber product of
$(H_{n}(\mathcal{H}), \nu, G_{\infty,\infty}(\mathcal{H}))$ and $(PI(I, G_{\infty,\infty}(\mathcal{H})),$$\mu,$ $G\infty,\infty(\mathcal{H}))$ by
$H_{n}(\mathcal{H})\cross_{G_{\infty,\infty}()}\mathcal{H}PI(I, G\infty,\infty(\mathcal{H}))$
We denote the natural projection
$p:H_{n}(\mathcal{H})\mathrm{x}_{G_{\infty,\infty}(\{)}\gamma PI(I, G\infty,\infty(\mathcal{H}))arrow G_{\infty,\infty}(\mathcal{H})$
.
$(H_{n}(\mathcal{H})\cross_{G_{\infty,\infty}(\mathcal{H})}PI(I, G_{\infty},\infty(\mathcal{H})), p, G_{\infty,\infty}(\mathcal{H}))$is a fibration, too.
Theorem 4.1 There is the following equivalence
of
fibrations:
(Rep$(\mathcal{O}_{n},$$\mathcal{H}),$
$\pi,$$G(\infty,\infty \mathcal{H})$ )
$\cong(H_{n}(\mathcal{H})\cross_{G_{\infty,\infty}(\mathcal{H})}PI(I, G_{\infty,\infty}(\mathcal{H})),$ $p,$ $G_{\infty,\infty}(\mathcal{H}))$.
Proof.
Define a map$\theta$ : Rep$(\mathcal{O}n’ \mathcal{H})arrow H_{n}(\mathcal{H})\mathrm{x}_{G_{\infty,\infty}()}\mathcal{H}PI(I, G\infty,\infty(\mathcal{H}))$,
$\theta(\phi)\equiv(\{\phi(siSj*))\}_{i,j}^{n}=1’\phi(_{S_{1})})$ $(\phi\in \mathrm{R}\mathrm{e}\mathrm{p}(\mathcal{O}_{n}, \mathcal{H}))$
where we identify a matrix unit and an element in $H_{n}(\mathcal{H})$.
Then we have
$\nu(\{\phi(S_{i}S_{j})*)\}in,j=1)=$ $\phi(s_{11}s^{*})$
$=$ $\phi(s_{1})\phi(s_{1})^{*}$
$=$ $\mu(\phi(S_{1}))$
.
Hence the image of$\theta$ is in $H_{n}(\mathcal{H})\cross_{G_{\infty,\infty}(H)}PI(I, G_{\infty,\infty}(\mathcal{H}))$
.
On
the other hand,$\theta^{-1}(\psi, v)=\{v, \psi(E_{21})v, \psi(E_{31})v, \ldots, \psi(En1)v\}$
for $((\psi, v)\in H_{n}(\mathcal{H})\mathrm{X}_{G_{\infty,\infty}(\mathcal{H})}PI(I, G\infty,\infty(\mathcal{H}))$ where we identify a Cuntz generator
and an element in Rep$(\mathcal{O}n’ \mathcal{H})$. Clearly, $(\theta, id_{G_{\infty},(\mathcal{H}})\infty)$ is a (set
$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}$)$\mathrm{f}\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$
isomorphism in Theorem 4.1. 1
4.3
General
case
In general, for two $\mathrm{C}^{*}$-algebra $A$ and $B,$ $\mathrm{H}\mathrm{o}\mathrm{m}(A, B)$ becomes the union of
homoge-neous space of the inner automorphism group $\mathrm{I}\mathrm{n}\mathrm{n}B$ of$B$ with the isotropy subgroup
$H_{\phi}$ defined by
$H_{\phi}\equiv$ $\mathrm{I}\mathrm{n}\mathrm{n}\mathcal{R}_{\phi}$
$\subset$ $\mathrm{I}\mathrm{n}\mathrm{n}B$,
$\mathcal{R}_{\phi}\equiv$ $\phi(A)’\cap B$
$=$ $\{b\in B:[a, b]=0, a\in\phi(A)\}$.
for $\phi\in \mathrm{H}\mathrm{o}\mathrm{m}(A, \beta)$. We have calculated concrete examples for $\mathrm{H}\mathrm{o}\mathrm{m}(A, B)$
.
We note that a similar theorem is known in a (commutative)mapping space
between differential manifold(section
13
in [4]).The Weyl form of the canonical quantization is a pair $(\phi, \psi)$ offaithful
represen-tation of $C(s^{1})$ on a separable infinite dimensional Hilbert space $\mathcal{H}$ such that they
other. That is, the canonical commutation relation can be regarded as a special
position oftwo points in $\mathrm{H}\mathrm{o}\mathrm{m}(C(s^{1}), \mathcal{L}(\mathcal{H}))$.
Since
Fourier transformation is agen-erator of$\mathrm{Z}_{4}$, it seems a rotation ofangle $90^{\mathrm{O}}$ in a 2-dimensional plane. Hence a pair
of points associated with CCRis seemedin a position of$90^{\mathrm{O}}$ in a some2-dimensional
plane in $\mathrm{H}\mathrm{o}\mathrm{m}(C(s^{1}), L(\mathcal{H})$ around center. When a pair $(\phi, \psi)$ takes a commutative
position, the algebra generated by $\phi(C(s^{1}))$ and $\psi(C(s^{1}))$ is isomorphic to $C(T^{2})$
.
Deformation of$C(T^{2})$ to $A_{\theta}$ can be treated in the space $\mathrm{H}_{\mathrm{o}\mathrm{m}}(c(s^{1}), \mathcal{L}(\mathcal{H}))$( or the quotient space of$\mathrm{H}_{\mathrm{o}\mathrm{m}}(c(s^{1}), \mathcal{L}(\mathcal{H})))$.
In this way, many examples of $\mathrm{C}^{*}$-algebra with some generators can be stated as
a position of of representations ofcommutative algebras.
5
A
categorical
extension
of Gelfand
transforma-tion
In the case (iii) in section 1, we show the following categorical reformulation of
Gelfand representation of commutative unital $\mathrm{C}^{*}$-algebras:
Fact 5.1 (Gelfand representation) Denote categories $\mathcal{L}C\mathcal{H},$ $C\mathcal{H}$ and $CC_{1}^{*}$ categories
of
locally compactHausdorff
spaces, compactHausdorff
spaces and unitalcommuta-tive $C^{*}$-algebras, respectively.
Then we have the following natural equivalence
$1_{CC_{1}^{*}}\cong$ $\mathcal{L}C\mathcal{H}_{\mathrm{C}^{\mathrm{O}(}}Cc^{*})_{\mathrm{C}}1$
’
$1_{C\mathcal{H}}\cong$ $(CC_{1}^{*})_{\mathrm{C}}0\mathcal{L}C\mathcal{H}\mathrm{c}|c\mathcal{H}$,
where $\mathcal{X}_{Y}$ is the contravariant principal representation
of
a category $\mathcal{X}=\mathcal{L}C\mathcal{H},Cc_{1}*$by an object $Y$ in $\mathcal{X}$ and $1_{\mathcal{X}}$ is the identity
functor
on $\mathcal{X}$.Remark that $\mathrm{C}$ is the set of all complex numbers. $\mathrm{C}$ is 1-dimensional
commu-tative unital $\mathrm{C}^{*}$-algebra and a locally compact Hausdorff space which is
homeo-morphic to 2-dimensional Euclid space $\mathrm{R}^{2}$. In the above fact, a covariant functor
Gel $\equiv \mathcal{L}C\mathcal{H}\mathrm{C}^{\mathrm{O}}(CC_{1}^{*})\mathrm{c}$ is a transformation of a category of $CC_{1}^{*}$. We call Gel the
Gelfand
transformation.
This reformulation gives us a question what is noncom-mutative version of Gel.
The reason of the success of this reformulation is caused by regarding the set
of pure states as the character of a commutative $\mathrm{C}^{*}$-algebra. Because of this
rea-son, we consider that a generalization of Gelfand transformation to non
commu-tative case may be suitable by principal representation of category $C_{1}^{*}$ of unital
(non $\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}$)$\mathrm{c}*$-algebras. We claim that a generalization of the category $C\mathcal{H}$ of compact Hausdorff spaces with respect to generalized Gelfand transformation
contravariant-covariant functor of a category $C_{1}^{*}$. For $A,$$B\in C_{1}^{*}$, define
$F_{2}(A, B)\equiv$ $F_{1}(A, \beta))\beta \mathrm{n}\mathrm{n}e$,
$F_{3}(A, \beta)\equiv$ $\{\mathrm{K}\mathrm{e}_{1}\mathrm{r}\phi:\phi\in F_{1}(A, B)\}$
$=$ $\{\mathrm{K}\mathrm{e}\mathrm{r}\phi:[\phi]\in F_{2}(A, \beta)\}$.
Let $r,$$s$ be natural projections between them
$F_{1}(A, B)rarrow F2(A,g)arrow F_{3}(SA, B)$
.
Proposition 5.1 (i) $F^{(1)}\equiv(F_{1}, r, F_{2})$ is a contravariant-covariant
bi-functor
from
$C_{1}^{*}$ to$\mathcal{F}\mathcal{I}\beta(1)$.
(ii) $F^{(2)}\equiv(F_{1}, r, F_{2}, s, F3)$ is a contravariant
functor
from
$C_{1}^{*}$ to$\mathcal{F}\mathcal{I}B^{(2}$).
where$\mathcal{F}\mathcal{I}B^{()}n$ is
the category
of
$n$-stepfibrations
which is a$n+1$-chain$(\{X_{i}\}^{n}i=0’\{p_{j}\}_{j=}^{n}1)$of
surjections between spaces:$X_{0}-p_{\int X1s}p$ $...-3pX_{n}$
.
We note that $F_{2}=F_{1}$ in commutative case. The notion of $F_{3}$ appears in the
theory of null ideal sequence. $F_{2}$ and $F_{3}$ are some kinds of generalization of spectrum
and primitive spectrum, respectively. Now we have not yet success to define good
topology for them. We must calculate examples of this categorical fibration more.
References
[1] R.Cirelli, A.Mani\‘a and L.Pizzocchero, A
functional
representationof
noncom-mutative $C^{*}$-algebras Rev.Math.Phys. Vol. 6, No.5 (1994)
675-697.
[2] K.Kawamura Structure theorem
of
the groupof
automorphismsof
$C^{*}$-algebras,math.$\mathrm{O}\mathrm{A}/9809109$
.
[3] K.Kawamura Serre-Swan theorem
for
non-commutative $C^{*}$-algebras,math.$\mathrm{O}\mathrm{A}/0002160$.
