We define a notion of depth for an inclusion of multimatrix algebras B ⊆ A based on a comparison of powers of the inductionrestriction table M (and its transpose matrix). The depth of the semisimple subalgebra B in the semisimple algebra A is the least positive integer n ≥ 2 for which M ≤ qM for some q ∈ Z+. We prove that a depth two subalgebra is a normal subalgebra, and conversely. As a corol...

‎Let $mathcal{A}_p$ be the mod $p$ Steenrod algebra‎, ‎where $p$ is an odd prime‎, ‎and let $mathcal{A}$ be the‎ subalgebra $mathcal{A}$ of $mathcal{A}_p$ generated by the Steenrod $p$th powers‎. ‎We generalize the $X$-basis in $mathcal{A}$ to $mathcal{A}_p$‎.

We study the subalgebra of the lattice vertex operator algebra V√ 2A2 consisting of the fixed points of an automorphism which is induced from an order 3 isometry of the root lattice A2. We classify the simple modules for the subalgebra. The rationality and the C2-cofiniteness are also established.

A well-known Peterson’s theorem says that the number of abelian ideals in a Borel subalgebra of a rank-r finite dimensional simple Lie algebra is exactly 2r . In this paper, we determine the dimensional distribution of abelian ideals in a Borel subalgebra of finite dimensional simple Lie algebras, which is a refinement of the Peterson’s theorem capturing more Lie algebra invariants.

We determine when there is a unique conditional expectation from a semifinite von Neumann algebra onto a singly-generated maximal abelian *-subalgebra. Our work extends the results of Kadison and Singer via new methods, notably the observation that a unique conditional expectation onto a singly-generated maximal abelian *-subalgebra must be normal.

A subclass of dynamical semigroups induced by the interaction of a quantum system with an environment is introduced. Such semigroups lead to the selection of a stable subalgebra of effective observables. The structure of this subalgebra is completely determined. ∗Supported by the KBN research grant No 2 P03B 086 16

Necessary and sufficient conditions will be found for quadruples of lattices to be isomorphic to lattices of weak, relative, strong subalgebras and initial segments, respectively, of one partial unary algebra. To this purpose we will start with a characterization of pairs of lattices that are weak and strong subalgebra lattices of one partial unary algebra, respectively. Next, we will describe ...

There is shown that there exists a complete, atomless, σ-centered Boolean algebra, which does not contain any regular countable subalgebra if and only if there exist a nowhere dense ultrafilter. Therefore the existence of such algebras is undecidable in ZFC. A subalgebra B of a Boolean algebra A is called regular whenever for every X ⊆ B, supB X = 1 implies supA X = 1; see e.g. Heindorf and Sha...

To calculate the entropy of a subalgebra or of a channel with respect to a state, one has to solve an intriguing optimalization problem. The latter is also the key part in the entanglement of formation concept, in which case the subalgebra is a subfactor. I consider some general properties, valid for these definitions in finite dimensions, and apply them to a maximal commutative subalgebra of a...

To calculate the entropy of a subalgebra or of a channel with respect to a state, one has to solve an intriguing optimalization problem. The latter is also the key part in the entanglement of formation concept, in which case the subalgebra is a subfactor. I consider some general properties, valid for these definitions in finite dimensions, and apply them to a maximal commutative subalgebra of a...