Let P be a pointed, closed convex cone in $\mathbb {R}^d$. We prove that for two pure isometric representations $V^{(1)}$ and $V^{(2)}$ of P, the associated CAR flows $\beta ^{V^{(1)}}$ and $\beta ^{V^{(2)}}$ are cocycle conjugate if and only if $V^{(1)}$ and $V^{(2)}$ are unitarily equivalent. We also give a complete description of pure isometric representations of P with commuting range projections that give rise to type I CAR flows. We show that such an isometric representation is completely reducible with each irreducible component being a pullback of the shift semigroup $\{S_t\}_{t \geq 0}$ on $L^2[0,\infty )$. We also compute the index and the gauge group of the associated CAR flows and show that the action of the gauge group on the set of normalized units need not be transitive.

Given a Fell bundle $\mathcal {B}=\{B_t\}_{t\in G}$ over a locally compact group G and a closed subgroup $H\subset G,$ we construct quotients $C^{*}_{H\uparrow \mathcal {B}}(\mathcal {B})$ and $C^{*}_{H\uparrow G}(\mathcal {B})$ of the full cross-sectional C*-algebra $C^{*}(\mathcal {B})$ analogous to Exel–Ng’s reduced algebras $C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B})\equiv C^{*}_{\{e\}\uparrow \mathcal {B}}(\mathcal {B})$ and $C^{*}_R(\mathcal {B})\equiv C^{*}_{\{e\}\uparrow G}(\mathcal {B}).$ An absorption principle, similar to Fell’s one, is used to give conditions on $\mathcal {B}$ and H (e.g., G discrete and $\mathcal {B}$ saturated, or H normal) ensuring $C^{*}_{H\uparrow \mathcal {B}}(\mathcal {B})=C^{*}_{H\uparrow G}(\mathcal {B}).$ The tools developed here enable us to show that if the normalizer of H is open in G and $\mathcal {B}_H:=\{B_t\}_{t\in H}$ is the reduction of $\mathcal {B}$ to $H,$ then $C^{*}(\mathcal {B}_H)=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B}_H)$ if and only if $C^{*}_{H\uparrow \mathcal {B}}(\mathcal {B})=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B});$ the last identification being implied by $C^{*}(\mathcal {B})=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B}).$ We also prove that if G is inner amenable and $C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B})\otimes _{\max } C^{*}_{\mathop {\mathrm {r}}}(G)=C^{*}_{\mathop {\mathrm { r}}}(\mathcal {B})\otimes C^{*}_{\mathop {\mathrm {r}}}(G),$ then $C^{*}(\mathcal {B})=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B}).$

Let P be a closed convex cone in $\mathbb{R}^d$ which is assumed to be spanning $\mathbb{R}^d$ and contains no line. In this article, we consider a family of CAR flows over P and study the decomposability of the associated product systems. We establish a necessary and sufficient condition for CAR flow to be decomposable. As a consequence, we show that there are uncountable many CAR flows which are cocycle conjugate to the corresponding CCR flows.

In this paper, we construct uncountably many examples of multiparameter CCR flows, which are not pullbacks of $1$-parameter CCR flows, with any given index. Moreover, the constructed CCR flows are type I in the sense that the associated product system is the smallest subsystem containing its units.

Let $P$ be a closed convex cone in $\mathbb{R}^{d}$ which is spanning, i.e., $P-P=\mathbb{R}^{d}$ and pointed, i.e., $P\,\cap -P=\{0\}$. Let $\unicode[STIX]{x1D6FC}:=\{{\unicode[STIX]{x1D6FC}_{x}\}}_{x\in P}$ be an $E_{0}$-semigroup over $P$ and let $E$ be the product system associated to $\unicode[STIX]{x1D6FC}$. We show that there exists a bijective correspondence between the units of $\unicode[STIX]{x1D6FC}$ and the units of $E$.

Let $G$ be a second countable locally compact Hausdorff topological group and $P$ be a closed subsemigroup of $G$ containing the identity element $e\in G$. Assume that the interior of $P$ is dense in $P$. Let $\unicode[STIX]{x1D6FC}=\{{\unicode[STIX]{x1D6FC}_{x}\}}_{x\in P}$ be a semigroup of unital normal $\ast$-endomorphisms of a von Neumann algebra $M$ with separable predual satisfying a natural measurability hypothesis. We show that $\unicode[STIX]{x1D6FC}$ is an $E_{0}$-semigroup over $P$ on $M$.

In this paper, we give a different proof of the fact that the odd dimensional quantum spheres are groupoid ${{C}^{*}}$ -algebras. We show that the ${{C}^{*}}$ -algebra $C\left( S_{q}^{2\ell +1} \right)$ is generated by an inverse semigroup $T$ of partial isometries. We show that the groupoid ${{\mathcal{G}}_{tight}}$ associated with the inverse semigroup $T$ by Exel is exactly the same as the groupoid considered by Sheu.

In this paper we generalise a result of Izuchi and Suárez (K. Izuchi and D. Suárez, Norm-closed invariant subspaces in L∞ and H∞, Glasgow Math. J. 46 (2004), 399–404) on the shift invariant subspaces of $L^\infty(\mathbb{T})$ to the non-commutative setting. Considering these subspaces as $C(\mathbb{T})$-modules contained in $L^\infty(\mathbb{T})$, we show that under some restrictions, a similar description can be given for the ${\mathfrak{B}}$-submodules of ${\mathfrak{A}}$, where ${\mathfrak{A}}$ is a C*-algebra and ${\mathfrak{B}}$ is a commutative C*-subalgebra of ${\mathfrak{A}}$. We use this to give a description of the $\mathbb{M}_n({\mathfrak{B}})$-submodules of $\mathbb{M}_n({\mathfrak{A}})$.

Suppose that $G$ is a second countable, locally compact Hausdorff groupoid with abelian stabiliser subgroups and a Haar system. We provide necessary and sufficient conditions for the groupoid ${C}^{\ast } $-algebra to have Hausdorff spectrum. In particular, we show that the spectrum of ${C}^{\ast } (G)$ is Hausdorff if and only if the stabilisers vary continuously with respect to the Fell topology, the orbit space ${G}^{(0)} / G$ is Hausdorff, and, given convergent sequences ${\chi }_{i} \rightarrow \chi $ and ${\gamma }_{i} \cdot {\chi }_{i} \rightarrow \omega $ in the dual stabiliser groupoid $\widehat{S}$ where the ${\gamma }_{i} \in G$ act via conjugation, if $\chi $ and $\omega $ are elements of the same fibre then $\chi = \omega $.

Recently, Daws introduced a notion of co-representation of abelian Hopf–von Neumann algebras on general reflexive Banach spaces. In this note, we show that this notion cannot be extended beyond subhomogeneous Hopf–von Neumann algebras. The key is our observation that, for a von Neumann algebra 𝔐 and a reflexive operator space E, the normal spatial tensor product is a Banach algebra if and only if 𝔐 is subhomogeneous or E is completely isomorphic to column Hilbert space.

We prove the equivalence of three definitions given by different comparison relations for infiniteness of positive elements in simple ${{C}^{*}}$ -algebras.

Suppose A is a unital C*-algebra and m:A → B(X) is unital bounded algebra homomorphism where B(X) is the algebra of all operators on a Banach space X. When X is a Hilbert space, a problem of Kadison [9] asks whether m is similar to a *-homomorphism. Haagerup [5] has shown that the answer is positive when m(A) has a cyclic vector or whenever m is completely bounded. We use this to show m(A) is reflexive (Alg Lat m(A) = m(A)−sot ) whenever X is a Hilbert space. Our main result is that whenever A is a separable GCR C*-algebra and X is a reflexive Banach space, then m(A) is reflexive.

Let G denote a plane domain bounded by finitely many closed, non-intersecting Jordan curves. We show the following refinement of the stable rank one property of : Suppose that for there exists δ > 0 such that . Then there exist such that

f + hg = exp (k).

Also we obtain a partial result for the algebra H∞(G).